High-dimensional asymptotics for percolation of Gaussian free field level sets

We consider the Gaussian free field on Z(d), d >= 3, and prove that the critical density for percolation of its level sets behaves like 1/d(1+o(1)) as d tends to infinity. Our proof gives the principal asymptotic behavior of the corresponding critical level h(*) (d). Moreover, it shows that a related parameter h(**) (d) >= h(*) (d) introduced by Rodriguez and Sznitman in [24] is in fact asymptotically equivalent to h(*) (d).